Zeroes of complete L-functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:22:14Z http://mathoverflow.net/feeds/question/52438 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52438/zeroes-of-complete-l-functions Zeroes of complete L-functions Sylvain JULIEN 2011-01-18T21:51:03Z 2011-01-19T18:15:25Z <p>Hello,</p> <p>Let $F$ and $G$ be two functions belonging in the Selberg class, $\Lambda_{F}$ and $\Lambda_{G}$ the complete L-functions associated to $F$ and $G$. I would like to know whether this assertion is true or not:</p> <p>"$\Lambda_{F}$ and $\Lambda_{G}$ have the same zeroes if and only if $F=G$ or $F=\overline{G}$."</p> <p>Thank you in advance.</p> http://mathoverflow.net/questions/52438/zeroes-of-complete-l-functions/52537#52537 Answer by Micah Milinovich for Zeroes of complete L-functions Micah Milinovich 2011-01-19T17:52:19Z 2011-01-19T18:15:25Z <p>Expanding on Stopple's comment above, I believe the following argument based on Landau's explicit formula answers the question.</p> <p>Here is a generalization of Landau's explicit formula for the zeros of the Riemann zeta-function which is exercise 8.4.8 in M. Ram Murty's book <em>Problems in Analytic Number Theory</em>: Let $F$ be in the Selberg class, $n>1$ be a positive integer, and $T>1$. Then $$\sum_{|\gamma|\leq T} n^\rho = -\frac{T}{\pi}\Lambda_F(n) + O( n^{3/2}\log T )$$ where $\rho=\beta+i\gamma$, $\beta>0$, runs over the non-trivial zeros of $F(s)$. Here the coefficients $\Lambda_F(n)$ are defined by $$-\frac{F'}{F}(s) = \sum_{n=1} \frac{\Lambda_F(n)}{n^s}.$$</p> <p>Now suppose that $F$ and $G$ are in the Selberg class and have the same zeros (with multiplicity). Then we deduce from Landau's formula that $$|\Lambda_F(n) - \Lambda_G(n)| \ll \frac{n^{3/2}\log T}{T}$$ for all $n>1$. Letting $T\rightarrow \infty$, it follows that $\Lambda_F(n) = \Lambda_G(n)$ for all $n>1$. This implies that $F=G$.</p>